Theorem 2.2 (Theorem 6.6 in [1]}) 
Let \((A,B,R)\)  be a Borel invariant. Suppose that \((\mathbb {P}_\alpha :\alpha <\omega _2)\)  is a sequence of Borel partial ordersIn an iteration, a Borel partial order is really a sequence of codes, interpreted in successive stages of the iteration. such that for each \(\alpha <\omega _2\) , \(\mathbb {P}_\alpha \)  is forcing equivalent to \(\mathcal {P}(\lbrace 0,1\rbrace )^+\times \mathbb {P}_\alpha \) .The non-triviality assumption that \(\mathbb {P}_\alpha \)  is equivalent to \(\mathcal {P}(\lbrace 0,1\rbrace )^+\times \mathbb {Q}_\alpha \)  is very mild; in nearly all of classical forcing notions, this can be accomplished by coding a condition in \(\mathcal {P}(\lbrace 0,1\rbrace )^+\)  into the “first coordinate” of the generic object. Let \(\mathbb {P}_{\omega _2}\)  be the countable support iteration of this sequence. If \(\mathbb {P}_{\omega _2}\)  is proper, then \(\mathbb {P}_{\omega _2}\)  forces \(\diamondsuit (A,B,R)\)  if and only if \(\mathbb {P}_{\omega _2}\)  forces \(\langle A,B,R\rangle \le \aleph _1\) .

